WASHINGTON (AP) — Senators moved toward resolving their feud over Republican filibusters of White House appointees on Tuesday, hoping to avoid a Senate rules change by Democrats that would worsen the partisanship already troubling the chamber.

Officials said both parties were discussing a plan to permit prompt confirmation for most of the contested nominees, including Tom Perez to head the Labor Department, Gina McCarthy to run the Environmental Protection Agency and Richard Cordray to lead the Consumer Financial Protection Bureau.

The Senate voted 71-29 to clear the way for eventual confirmation of Cordray, whom President Barack Obama installed when the Senate was in recess, angering Republicans.

Senate Majority Leader Harry Reid expressed confidence that a broader deal was within reach, even if it would leave open the possibility of future filibusters of Obama’s executive nominees. Reid had been threatening to change the rules, to bar such filibusters.

Senators are not questioning the ability to keep using filibusters -- in which 41 of the 100 senators can block action -- on legislation and judicial nominees, who seek lifetime appointments.

Under the current proposal, Obama would drop efforts to win confirmation for two members of the National Labor Relations Board and name two replacements who would receive speedy consideration. Richard Griffin and Sharon Block were originally named to their posts as recess appointees, meaning they bypassed confirmation. An appeals court has since ruled their appointments were invalid, and the Supreme Court has agreed to review the case.

The proposed agreement would not resolve deep partisan divisions over the future use of filibusters to block a president’s executive nominees.

‘‘It is a compromise, and I think we get what we want, they get what they want. Not a bad deal,’’ Reid said on the Senate floor.

The developments unfolded the morning after a closed-door meeting of nearly all 100 senators, eager to avoid a rules change that could poison relations between the two parties.

If ratified, the deal would mark a retreat by Reid from his insistence on Monday that Republicans promise not to filibuster future executive nominees. Republican leader Mitch McConnell had privately offered to clear the way for the currently contested nominees -- providing Block and Griffin were replaced -- officials in both parties said. That’s largely the deal Democrats agreed to on Tuesday.

McCain told reporters it was ‘‘probably the hardest thing I've been involved in.’’ Noting the Senate recently passed a bipartisan immigration bill, he said, ‘‘Maybe we can show more momentum toward bipartisanship. Is it a panacea? No, but I think it’s an important step forward.’’

Democrats acknowledged that a rules change probably would have prompted Republicans to retaliate by doing even more to reduce the minority party’s rights if the GOP regained control of the Senate. That could happen as early 18 months from now, after the 2014 elections.

‘‘It’s a decision that, if they actually go through with it, they will live to regret,’’ McConnell has said.

Unlike the 435-member House, the Senate has a long and bumpy tradition of granting rights to minority-party members. The most powerful tool is the filibuster, which can kill a measure by using endless debate to prevent a yes-or-no vote.

The mere promise of a filibuster can block Senate action on almost anything unless 60 of the 100 senators vote to overcome it. Filibuster-proof majorities are rare, and Republicans now hold 46 Senate seats.

Both parties have accelerated their use of the filibuster threat in recent times. Since Obama took office in January 2009, Republicans have threatened filibusters repeatedly, infuriating Democrats.

Reid said Lyndon B. Johnson faced one filibuster during his six years as Senate majority leader. In the same length of time as majority leader, Reid said he has faced 413 threatened filibusters. The tactic, he said, blocks action on routine matters that Congress once handled fairly easily.

White House spokesman Jay Carney told reporters the Senate ‘‘needs to confirm this president’s nominees in a timely and efficient manner.’’ That will be true, he said, ‘‘for the next president, and the next president after that. This has become ridiculous.’’

Asked Monday if Obama worries that a filibuster rule change would make the Senate even more dysfunctional, Carney said, ‘‘Well, it boggles the mind how they would achieve that.’’

This notion that things can’t get much worse in the often stalemated Senate seems to have convinced numerous senators and interest groups in recent months that there was little risk in talking about changing traditions to end at least some of the logjams.

Associated Press writers Alan Fram and David Espo contributed to this report.

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